pith. sign in

arxiv: 2603.28070 · v2 · submitted 2026-03-30 · 🧮 math.AP

Incompressible Euler equations in 3D bounded domains in a critical space

Tsukasa Iwabuchi , Hideo Kozono This is my paper

Pith reviewed 2026-05-14 22:22 UTC · model grok-4.3

classification 🧮 math.AP
keywords Euler equationsbounded domainsBesov spaceStokes operatorvanishing viscositylocal existencestrong solutionsincompressible flow
0
0 comments X

The pith

Unique local strong solutions exist for the 3D incompressible Euler equations in bounded domains for initial data in the critical Besov space B^{5/2}_{2,1}(A).

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves the existence and uniqueness of local-in-time strong solutions to the three-dimensional incompressible Euler equations inside smooth bounded domains. The initial velocity is required to lie in a critical Besov space constructed from the Stokes operator with Neumann boundary conditions. The argument proceeds by approximating the Euler equations with the Navier-Stokes equations and passing to the zero-viscosity limit. Uniform bounds independent of viscosity are obtained through commutator estimates that control the nonlinear term in the chosen function space. This result supplies a functional-analytic setting in which the ideal fluid equations are locally well-posed near the boundary.

Core claim

For the three-dimensional incompressible Euler equations in a bounded domain Ω with smooth boundary, there exists T > 0 and a unique strong solution u on [0, T) whenever the initial datum u_0 belongs to the critical space B^{5/2}_{2,1}(A), where A is the Stokes operator subject to the Neumann boundary condition. The solution is constructed as the limit of solutions to the Navier-Stokes equations as the viscosity parameter tends to zero.

What carries the argument

The critical Besov space B^{5/2}_{2,1}(A) generated by the Stokes operator A, which encodes the boundary condition and provides the precise scaling for local well-posedness of the Euler system.

If this is right

  • The solution remains in the same Besov space for a positive time interval.
  • The vanishing viscosity limit yields a strong solution to the Euler equations.
  • Energy estimates are uniform in the viscosity parameter.
  • Local existence holds without additional compatibility conditions on the initial data beyond membership in the space.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The result opens the possibility of studying boundary effects on singularity formation in ideal fluids.
  • Similar techniques might apply to other inviscid equations such as the Euler-Poisson system in domains.
  • Global regularity questions could be addressed by combining this local theory with conserved quantities or symmetries.

Load-bearing premise

The commutator estimates between the Stokes operator and the nonlinear term must produce bounds that stay uniform as viscosity vanishes.

What would settle it

A concrete initial datum in B^{5/2}_{2,1}(A) for which either no solution exists or the solution ceases to be strong in arbitrarily short time would disprove the theorem.

read the original abstract

We consider the 3D incompressible Euler equations in bounded domains $\Omega$ with smooth boundary $\partial\Omega$. Based on the paper by Iwabuchi, Matsuyama and Taniguchi (2019), we define the Besov space $B^s_{p, q}(A)$ by means of the Stokes operator $A$ with the Neumann boundary condition on $\partial\Omega$, and prove unique local existence theorem of strong solution for the initial data in the critical Besov space $B^{\frac52}_{2, 1}(A)$. Our proof relies on the method of vanishing viscosity. The commutator estimate plays an essential role for derivation of energy bounds which hold uniformly with respect to viscosity constants.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript proves a unique local existence theorem for strong solutions of the 3D incompressible Euler equations in a bounded domain Ω with smooth boundary, for initial data in the critical Besov space B^{5/2}_{2,1}(A) defined via the Stokes operator A with Neumann boundary conditions. The argument proceeds by vanishing-viscosity approximation: solutions u_ν to the Navier-Stokes system are constructed and the limit ν→0 is taken, with the key step being an a priori estimate in B^{5/2}_{2,1}(A) that is independent of ν and obtained via a commutator estimate applied to the nonlinear term.

Significance. If the uniform bounds are established, the result would extend local well-posedness for the Euler equations to critical Besov spaces in bounded domains, a setting where boundary conditions must be respected. Defining the Besov space through the Stokes operator is a natural adaptation that aligns with the Neumann boundary condition. The vanishing-viscosity approach combined with commutator estimates is standard, and a successful verification here would supply a reusable template for related boundary-value problems.

major comments (1)
  1. [Section deriving the a priori estimates via the commutator estimate] The derivation of the ν-uniform bound in B^{5/2}_{2,1}(A) (the central step for choosing the existence time independent of viscosity) invokes the commutator estimate from Iwabuchi-Matsuyama-Taniguchi (2019) after applying A^{5/4} to the nonlinear term. In the bounded-domain setting the commutator [A^s, u·∇] generates boundary integrals absent on R^3 or T^3; the manuscript does not supply an explicit estimate showing these integrals remain controlled by constants independent of ν when the domain is only C^∞ and the data lie in B^{5/2}_{2,1}(A). Without this verification the passage to the limit cannot be justified.
minor comments (1)
  1. [Abstract] The abstract states that the commutator estimate 'plays an essential role' but does not indicate whether the 2019 reference already covers bounded domains or whether additional boundary analysis is supplied in the present work; a single clarifying sentence would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The major concern regarding the handling of boundary integrals in the commutator estimate is well-taken, and we address it below with a commitment to revise the paper accordingly.

read point-by-point responses

  1. Referee: The derivation of the ν-uniform bound in B^{5/2}_{2,1}(A) (the central step for choosing the existence time independent of viscosity) invokes the commutator estimate from Iwabuchi-Matsuyama-Taniguchi (2019) after applying A^{5/4} to the nonlinear term. In the bounded-domain setting the commutator [A^s, u·∇] generates boundary integrals absent on R^3 or T^3; the manuscript does not supply an explicit estimate showing these integrals remain controlled by constants independent of ν when the domain is only C^∞ and the data lie in B^{5/2}_{2,1}(A). Without this verification the passage to the limit cannot be justified.

    Authors: We agree that an explicit verification of the boundary integrals is necessary for rigor in the bounded-domain setting. The commutator estimate from Iwabuchi-Matsuyama-Taniguchi (2019) is adapted using the spectral properties of the Stokes operator A with Neumann boundary conditions. In the revision, we will insert a dedicated subsection deriving the bound on the boundary terms via integration by parts, trace inequalities in Besov spaces, and the C^∞ regularity of ∂Ω, confirming that they are absorbed into the main terms with constants independent of ν. This ensures the a priori estimate remains uniform and justifies the vanishing-viscosity limit. revision: yes

Circularity Check

0 steps flagged

Minor self-citation to 2019 commutator estimate; central existence result remains independent

full rationale

The paper defines B^s_{p,q}(A) via the Stokes operator following the 2019 Iwabuchi-Matsuyama-Taniguchi reference and invokes their commutator estimate to obtain viscosity-uniform energy bounds for the vanishing-viscosity limit from Navier-Stokes to Euler. The local-existence theorem in B^{5/2}_{2,1}(A) is then derived by standard approximation and passage to the limit. No quantity is defined in terms of the target existence time or solution, no parameter is fitted to a subset of data and renamed a prediction, and the commutator itself is not re-derived inside this manuscript. The cited estimate supplies external analytic input rather than reducing the main claim to a self-referential loop.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard functional-analytic properties of the Stokes operator in smooth bounded domains and on commutator estimates for the bilinear term; no new free parameters or invented entities are introduced.

axioms (2)
  • standard math The Stokes operator A with Neumann boundary condition generates an analytic semigroup on suitable spaces in smooth bounded domains
    Invoked to define the Besov space B^s_{p,q}(A) and to obtain the necessary regularity for the linear part.
  • domain assumption Commutator estimates hold uniformly with respect to the viscosity parameter for the nonlinear term in the Euler equations
    Stated as essential for obtaining energy bounds independent of viscosity in the vanishing-viscosity limit.

pith-pipeline@v0.9.0 · 5412 in / 1456 out tokens · 94408 ms · 2026-05-14T22:22:38.432792+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages

  1. [1]

    Bahouri, J.-Y

    H. Bahouri, J.-Y. Chemin, and R. Danchin,Fourier analysis and nonlinear partial differential equa- tions, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 343, Springer, Heidelberg, 2011

    work page 2011

  2. [2]

    J. P. Bourgougnon and H. Brezis,Remarks on the Euler equation, J. Functional Anal.15(1974), no. 4, 341–363

    work page 1974

  3. [3]

    Bourgain and D

    J. Bourgain and D. Li,Strong ill-posedness of the 3D incompressible Euler equation in borderline spaces, Int. Math. Res. Not. IMRN16(2021), 12155–12264

    work page 2021

  4. [4]

    Chae,On the well-posedness of the Euler equations in the Triebel– Lizorkin spaces., Comm

    D. Chae,On the well-posedness of the Euler equations in the Triebel– Lizorkin spaces., Comm. Pure Appl. Math.55(2002), no. 5, 654—678

    work page 2002

  5. [5]

    Farwig and T

    R. Farwig and T. Iwabuchi,Sobolev spaces on arbitrary domains and semigroups generated by the fractional Laplacian, Bull. Sci. Math.193(2024), Paper No. 103440, 26

    work page 2024

  6. [6]

    Giga and T

    Y. Giga and T. Miyakawa,Solutions inL r of the Navier-Stokes initial value problem, Arch. Rational Mech. Anal.89(1985), no. 3, 267–281

    work page 1985

  7. [7]

    Iwabuchi,The semigroup generated by the Dirichlet Laplacian of fractional order, Anal

    T. Iwabuchi,The semigroup generated by the Dirichlet Laplacian of fractional order, Anal. PDE11(2018), no. 3, 683–703

    work page 2018

  8. [8]

    Iwabuchi and H

    T. Iwabuchi and H. Kozono,Besov space approach to the Navier-Stokes equations with the Neumann boundary condition in bounded domains, arXiv:2603.05779

    work page arXiv

  9. [9]

    Iwabuchi, T

    T. Iwabuchi, T. Matsuyama, and K. Taniguchi,Besov spaces on open sets, Bull. Sci. Math.152(2019), 93–149

    work page 2019

  10. [10]

    ,Boundedness of spectral multipliers for Schr¨ odinger operators on open sets, Rev. Mat. Iberoam.34 (2018), no. 3, 1277–1322

    work page 2018

  11. [11]

    Kato and C

    T. Kato and C. Y. Lai,Nonlinear evolution equations and Euler flows, J. Functional Anal.56(1984), 15–28. 26 TSUKASA IWABUCHI AND HIDEO KOZONO

    work page 1984

  12. [12]

    Kato and G

    T. Kato and G. Ponce,Commutator estimates and the Euler and Navier-Stokes equations, Commu. Pure Appl. Math.41(1988), 891–907

    work page 1988

  13. [13]

    Kozono and T

    H. Kozono and T. Yanagisawa,L r-variational inequality for vector fields and the Helmholtz-Weyl decompo- sition in bounded domains, Indiana Univ. Math. J.58(2009), no. 4, 1853–1920

    work page 2009

  14. [14]

    Kozono, S

    H. Kozono, S. Shimizu, and T. Yanagisawa,Stability of harmonic vector fields as an equilibrium of the 3D MHD equations in bounded domains with arbitrary geometry, Math. Ann393(2025), 923–952

    work page 2025

  15. [15]

    Nirenberg,On elliptic partial differential equations, Ann

    L. Nirenberg,On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3)13(1959), 115–162. MR0109940

    work page 1959

  16. [16]

    H. C. Pak and Y. J. Park,OExistence of solution for the Euler equations in a critical BesovB 1 ∞,1(Rn), Comm. Partial Differential Equations29(2004), 1149–1166

    work page 2004

  17. [17]

    Miyakawa,TheL p approach to the Navier-Stokes equations with the Neumann boundary condition, Hi- roshima Math

    T. Miyakawa,TheL p approach to the Navier-Stokes equations with the Neumann boundary condition, Hi- roshima Math. J.10(1980), no. 3, 517–537. MR0594132

    work page 1980

  18. [18]

    Teman,On the Euler equations of incompressible perfect fluids, J

    R. Teman,On the Euler equations of incompressible perfect fluids, J. Functional Anal.20(1975), 32–43

    work page 1975

  19. [19]

    Vishik,Incompressible flows of an ideal fluid with vorticity in borderline spaces of Besov type, Ann

    M. Vishik,Incompressible flows of an ideal fluid with vorticity in borderline spaces of Besov type, Ann. Sci. ´Ecole Norm. Sup. (4)32(1999), no. 6, 769-–812

    work page 1999

  20. [20]

    Yu and F

    Y. Yu and F. Liu,Ill-posedness for the Navier-Stokes and Euler equations in Besov spaces, Appl. Math.69 (2024), no. 6, 757-–767. (T. Iwabuchi)Mathematical Institute, Tohoku University, Sendai 980–8587, Japan, Email address, T. Iwabuchi:t-iwabuchi@tohoku.ac.jp (H. Kozono)Department of Mathematics, F aculty of Science and Engineering, W aseda Univer- sity, ...

    work page 2024